Complementary Angles

In geometry, identifying geometric properties of shapes such as lines and angles is an essential skill that you need to acquire to understand more profound concepts such as determining and finding the measures of angles.

This time, we will familiarize ourselves with another unique pair of angles called complementary angles.

Are you thrilled to learn about this type of angle pair? Get ready as we are going to go in-depth in exploring the world of complementary angles!

…but first, let’s define an angle before we go deeper into understanding complementary angles.

Complementary Angles

We already know that an angle is a figure formed by joining two rays at a common endpoint. The two rays are the sides or arms of an angle, and their common endpoint is called the vertex.

So, what exactly are complementary angles?

Complementary angles are a pair of angles that, when combined, sum up to 90 degrees.

You can imagine them as two puzzle pieces that fit together to form a 90-degree angle. When talking about complementary angles, it is important to keep in mind that they always appear in twos.

We can say that one angle is the complement of the other or that one angle is complementary to another.

If an angle measures 90 degrees, it is called a right angle. Since right angles do not require another piece of the puzzle to complete the 90-degree angle, they do not have complements and cannot also be called a complement of their own.

Three or more angles cannot also be called complementary angles even if their measures add up to 90 degrees because, by definition, complementary angles are always a pair.

As a rule, the measures of complementary angles are always positive.

Notice that complementary angles are always acute because each of the complements must measure less than 90 degrees to add up to 90 degrees.

In the illustration below, the sum of $$60^\circ$$ and $$30^\circ$$ is $$90^\circ$$ , forming a right angle. Thus, we can say that $$60^\circ$$ is a complement of $$30^\circ$$ and vice versa.

Similarly, the sum of two $$45^\circ$$ angles is which also forms a right angle. Hence, the complement of $$45^\circ$$ is also $$45^\circ$$ .

You might have heard of or used the word “complimentary” before, and you might think that it has nothing to do about adding up to 90 degrees.

Notice that the spelling of the word “complimentary” is spelled slightly different from what we are talking about here, which is “complementary.” So, be careful with the spelling, okay?

Did you know that…

The word “complementary” is derived from two Latin words, “complere,” which means “complete,” and “plere,” which means “fill?” The actual meaning of the word “complementary” is “the combination of objects or things in such a way that they complete each other or enhance the qualities of one another.”

Isn’t it cool to know the origin of the word complementary?

Types of Complementary Angles

There are two types of complementary angles – the adjacent complementary angles and the non-adjacent complementary angles.

Let’s take a look at how these two types differ from one another.

Adjacent Complementary Angles

These are complementary angles that share a common vertex and a common side. Below are illustrations of adjacent complementary angles.

Non-adjacent Complementary Angles

These are complementary angles that are not adjacent to each other. Below are illustrations of non-adjacent complementary angles.

Finding the Complement of an Angle

At some point in time, we may encounter situations that may require us to find the complement of an angle. Are you thrilled to know how to do it?

Here’s a simple method of determining the complement of a specific angle!

Since complementary angles add up to 90 degrees, we can determine the complement of an angle by subtracting the angle’s measure from $$90^\circ$$ .

For example, let us determine the complement of an angle whose measure is $$36^\circ$$ . To do this, we must subtract 36 from 90. So, 90 – 36 = 54.

Therefore, the complement of an angle whose measure is $$36^\circ$$ is an angle whose measure is $$54^\circ$$.

What a simple thing to do! Now you know how to find the complement of a specific angle.

Complementary angles also have some interesting characteristics and features that we’ll look at in the next section.

Properties of Complementary Angles

To fully understand complementary angles, you must keep in mind the following concepts that include some important characteristics and features:

• Complementary angles are those whose sum of the measures is equal to 90 degrees.
• If two angles are complementary, we call each angle “complement” or “complement angle” of the other angle.
• Complementary angles can be either adjacent or non-adjacent.
• Even if the sum of three or more angles is 90 degrees, they cannot be considered complementary angles. Complementary angles always appear in pairs.
• Two right angles or two obtuse angles can’t complement one another.
• A pair of complementary angles is always acute, but not all pairs of acute angles are complementary.

Complementary Angle Theorem

The complementary angle theorem states that if two angles are complementary to the same angle, then they are congruent.

But how did this theorem exist? Let’s find out by proving.

Suppose we have three angles, namely $$\angle1$$, $$\angle2$$, and $$\angle3$$, as shown in the figure below, and $$\angle2$$ and $$\angle3$$ are complementary to $$\angle1$$ . We need to prove that $$\angle2$$ and $$\angle3$$ are congruent.

Given that $$\angle2$$ and $$\angle3$$ are complementary to $$\angle1$$, we can say by the definition of complementary angles that $$\text{m ∠1 + m ∠2 = 90°}$$ and $$\text{m ∠1 + m ∠3 = 90°}$$.

By Subtraction Property of Equality, we can rewrite the previous equations as $$\text{m ∠2 = 90°- m ∠1}$$ and $$\mathrm m\;\angle3\;=\;90^\circ\;-\;\mathrm m\;\angle1$$.

Since the left-hand side of both equations is equal then, $$m\;\angle2\;=\;m\;\angle3$$.

Given that the measures of $$\angle2$$ and $$\angle3$$ are equal, then they are congruent angles.

Therefore, we have proven the theorem.

Solving Problems Involving Complementary Angles

Let us now try to solve some problems related to what we have just learned about complementary angles.

Problem 1

Find the measure of angle y in the figure.

In the given figure, y and $$34^\circ$$ forms a right angle which means that they are complementary angles. Hence, their sum is $$90^\circ$$.
So, we can write it as $$\mathrm y\;+\;34^\circ\;=\;90^\circ$$. Solving for the value of x, we can rewrite the equation using as $$y\;=\;90^\circ\;-\;34^\circ$$. Simplifying further, we’ll have $$y\;=\;56^\circ$$.
Therefore, angle y measures $$56^\circ$$.

Now, let’s solve another problem.

Problem 2

Suppose $$\angle A$$ and $$\angle B$$ are complementary angles. If $$\angle B$$ is twice as large as $$\angle A$$, find the measure of each angle.

This looks challenging but, it is really quite simple to solve.

Since $$\angle A$$ and $$\angle B$$ are complementary angles, then we know that that their sum is $$90^\circ$$. Writing it in a mathematical sentence, we will have $$m\;\angle A\;+\;m\;\angle B\;=\;90^\circ$$.

Given that $$\angle B$$ is twice as large as $$\angle A$$, we can say that $$m\;\angle A\;=\;x$$ and $$m\;\angle B\;=\;2x$$.

Substituting the value of $$m\;\angle A$$ and $$m\;\angle B$$, to our first equation, we’ll have, $$x\;+\;2x\;=\;90^\circ$$. By simplifying the equation, we will arrive at $$x\;=\;30^\circ$$. Since $$m\;\angle A\;=\;x$$, then by substitution, $$m\;\angle A\;=\;30^\circ$$.

Let us now substitute the value of x in $$m\;\angle B\;=\;2x$$ to get the measure of $$\angle B$$ . So, $$m\;\angle B\;=\;2(30^\circ)\;=\;60^\circ$$.

Therefore, the measure of $$\angle A$$ is $$30^\circ$$ and the measure of $$\angle B$$ is $$60^\circ$$ .

Complementary Angles in Real World

Complementary angles are not just a term used in mathematics. You can also encounter this significant pair of angles in real-life.

For example, when you slice a rectangular-shaped bread along the diagonal, you will have two right triangles, each with a pair of complementary angles.

More so, when the clock’s hour hand and minute hand form a right angle, say 3 o’clock, whenever the second-hand passes between 12 and 3, it forms complementary angles.

Another fine thing about complementary angles is that it is used to create designs. Having the two boards meet with perfect 45-degree cuts leaves the corner at 90 degrees. These diagonal cuts can be seen in the corners of picture frames, which add a better aesthetic feature.

Can you tell other things that exhibit complementary angles?

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Now that you have come to know a lot of things about complementary angles, are you ready to practice and apply your knowledge about it?